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Description
Title: | Annular breadth of hinges & hinge exit paths of annuli |
Author(s): | Tichenor, Scott R. |
Director of Research: | Alexander, Stephanie |
Doctoral Committee Chair(s): | Reznick, Bruce |
Doctoral Committee Member(s): | Wetzel, John E; Bishop, Richard |
Department / Program: | Mathematics |
Discipline: | Mathematics |
Degree Granting Institution: | University of Illinois at Urbana-Champaign |
Degree: | Ph.D. |
Genre: | Dissertation |
Subject(s): | exit path
escape path width annulus polygonal arc breadth Wetzel broadworm Voronoi Rivlin |
Abstract: | Given a compact set $\textsf{S}\subset\mathds{R}^2$, we define the annular width function for $\textsf{S}$, denoted $w(E)$, as the width of the annulus of support of $\textsf{S}$ centered at $E\in\overline{\mathds{R}^2}$, where $\overline{\mathds{R}^2}$ is an extension of the real plane $\mathds{R}^2$. The annular breadth of $\textsf{S}$ is defined as the absolute minimum of $w(E)$. We find the $2$-segment polygonal arc with the greatest annular breadth. For a given set $\textsf{S}\subset\mathds{R}^2$, an exit path of $\textsf{S}$ is a curve that cannot be covered by the interior of $\textsf{S}$. Given an annulus, we find its shortest $1$- or $2$-segment polygonal arc exit path(s). Bezdek and Connelly provided a lengthy and technically demanding proof that \emph{All orbiforms of width} $1$ \emph{are translation covers of the set of closed planar curves of length} $2$ \emph{or less}. We provide a short and simple proof that \emph{All orbiforms of width} $1$ \emph{are covers of the set of all planar curves of length} $1$ \emph{or less}. We also provide a proof that \emph{The Reuleaux triangle of width} $1$ \emph{is a cover of the set of all closed curves of length} $2$ using a recent of Wichiramala. |
Issue Date: | 2019-07-03 |
Type: | Text |
URI: | http://hdl.handle.net/2142/105618 |
Rights Information: | Copyright 2019 by Scott R. Tichenor. All rights reserved. |
Date Available in IDEALS: | 2019-11-26 |
Date Deposited: | 2019-08 |
This item appears in the following Collection(s)
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Dissertations and Theses - Mathematics
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Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois